(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
proper(c) → ok(c)
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
(1) QTRS Reverse (EQUIVALENT transformation)
We applied the QTRS Reverse Processor [REVERSE].
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
c'(active(x)) → c'(g(f(mark(x))))
g(f(active(x))) → g(mark(x))
c'(proper(x)) → c'(ok(x))
f(proper(x)) → proper(f(x))
g(proper(x)) → proper(g(x))
ok(f(x)) → f(ok(x))
ok(g(x)) → g(ok(x))
mark(top(x)) → proper(top(x))
ok(top(x)) → active(top(x))
Q is empty.
(3) QTRS Reverse (EQUIVALENT transformation)
We applied the QTRS Reverse Processor [REVERSE].
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
c'(active(x)) → c'(g(f(mark(x))))
g(f(active(x))) → g(mark(x))
c'(proper(x)) → c'(ok(x))
f(proper(x)) → proper(f(x))
g(proper(x)) → proper(g(x))
ok(f(x)) → f(ok(x))
ok(g(x)) → g(ok(x))
mark(top(x)) → proper(top(x))
ok(top(x)) → active(top(x))
Q is empty.
(5) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
ACTIVE(c) → F(g(c))
ACTIVE(c) → G(c)
PROPER(f(X)) → F(proper(X))
PROPER(f(X)) → PROPER(X)
PROPER(g(X)) → G(proper(X))
PROPER(g(X)) → PROPER(X)
F(ok(X)) → F(X)
G(ok(X)) → G(X)
TOP(mark(X)) → TOP(proper(X))
TOP(mark(X)) → PROPER(X)
TOP(ok(X)) → TOP(active(X))
TOP(ok(X)) → ACTIVE(X)
The TRS R consists of the following rules:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
proper(c) → ok(c)
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 6 less nodes.
(8) Complex Obligation (AND)
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(ok(X)) → G(X)
The TRS R consists of the following rules:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
proper(c) → ok(c)
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(10) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(ok(X)) → G(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(12) UsableRulesReductionPairsProof (EQUIVALENT transformation)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
G(ok(X)) → G(X)
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(G(x1)) = 2·x1
POL(ok(x1)) = 2·x1
(13) Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(14) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(15) TRUE
(16) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(17) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(ok(X)) → G(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(18) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(ok(X)) → F(X)
The TRS R consists of the following rules:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
proper(c) → ok(c)
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(19) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(20) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(ok(X)) → F(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(21) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(22) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(ok(X)) → F(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(23) UsableRulesReductionPairsProof (EQUIVALENT transformation)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
F(ok(X)) → F(X)
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(F(x1)) = 2·x1
POL(ok(x1)) = 2·x1
(24) Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(25) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(26) TRUE
(27) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PROPER(g(X)) → PROPER(X)
PROPER(f(X)) → PROPER(X)
The TRS R consists of the following rules:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
proper(c) → ok(c)
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(28) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(29) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PROPER(g(X)) → PROPER(X)
PROPER(f(X)) → PROPER(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(30) UsableRulesReductionPairsProof (EQUIVALENT transformation)
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
PROPER(g(X)) → PROPER(X)
PROPER(f(X)) → PROPER(X)
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [POLO]:
POL(PROPER(x1)) = 2·x1
POL(f(x1)) = 2·x1
POL(g(x1)) = 2·x1
(31) Obligation:
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(32) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(33) TRUE
(34) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(35) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PROPER(g(X)) → PROPER(X)
PROPER(f(X)) → PROPER(X)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(36) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(X))
The TRS R consists of the following rules:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
proper(c) → ok(c)
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(37) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(38) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(X))
The TRS R consists of the following rules:
proper(c) → ok(c)
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
g(ok(X)) → ok(g(X))
f(ok(X)) → ok(f(X))
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(39) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
TOP(mark(X)) → TOP(proper(X))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
The following usable rules [FROCOS05] were oriented:
proper(f(X)) → f(proper(X))
proper(c) → ok(c)
active(c) → mark(f(g(c)))
f(ok(X)) → ok(f(X))
g(ok(X)) → ok(g(X))
proper(g(X)) → g(proper(X))
active(f(g(X))) → mark(g(X))
(40) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(X)) → TOP(active(X))
The TRS R consists of the following rules:
proper(c) → ok(c)
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
g(ok(X)) → ok(g(X))
f(ok(X)) → ok(f(X))
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(41) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(42) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(X)) → TOP(active(X))
The TRS R consists of the following rules:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
g(ok(X)) → ok(g(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(43) MRRProof (EQUIVALENT transformation)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
TOP(ok(X)) → TOP(active(X))
Strictly oriented rules of the TRS R:
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
Used ordering: Polynomial interpretation [POLO]:
POL(TOP(x1)) = x1
POL(active(x1)) = 1 + x1
POL(c) = 0
POL(f(x1)) = x1
POL(g(x1)) = x1
POL(mark(x1)) = x1
POL(ok(x1)) = 2 + x1
(44) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
g(ok(X)) → ok(g(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(45) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(46) TRUE
(47) UsableRulesProof (EQUIVALENT transformation)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
(48) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(X))
The TRS R consists of the following rules:
proper(c) → ok(c)
proper(f(X)) → f(proper(X))
proper(g(X)) → g(proper(X))
g(ok(X)) → ok(g(X))
f(ok(X)) → ok(f(X))
active(c) → mark(f(g(c)))
active(f(g(X))) → mark(g(X))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.